Stagger tuned amplifier



Oct. 19, 1948. H. WALLMAN 2,451,893 r STAGGER TUNED AMPLIFIER Fild Aug. 50, 1945 FIG. I

0 IO F FIG. 2

STAGE GAIN X I OVER STAGGERINGI DOUBLE OVERALL BANDWIDTH HUMPED SELECTIVITY CURVE STAGGERED PAIR SYNGHRONOUS SINGLE TUNED [SEPARATION OF PEAKS o .5 L0 BANDWIDTH OF EACH STAGE ATTORNEY.

Patented Oct. 19, 1948 UNITED STATES PAT ENT OFFICE f I Q ;2",451,893-1 TUNED AMPLIFIER Henry Gambridge, Mass, assignor, by mesne assignments, to- Unitedz- States of; America as represented by theSecr-etary oflWar Application August 30, 1945 Serial No. 613,605

The present invention relates generally to a c0mmunicati cns method and more particularly to a method? of tuningsuccessivestages of a single tuned electronic amplifier.

In an electronic amplifier stage using agiiven type of vacuum tube" and coupling circuit; the gain and bandwidth-"are dependent in: such a manner that it one is increased; the. other is: decreased, the'prodhct-otthe two beingisubstantiall y constant. It is: alsol well known that it several amplifier stagesare connected cascade, the re sulting product of overall bandwidth and stage gain of the combination is much less: than; that for a single stage: it special case: of this is a wnchronous single-tuned IF amplifier; that is, one whose stages are all tuned to the same centertrequency-i 'l he producttof bandwidth and stage of a receiver in which such-"amamplifier' is used decreases rapidlsc asmhe'nnmbenofi IE"sta;gesincreases. T

In: certain types: or; radio object; locating systcms, is necessary toiadapt a receiver. to) pass: pulses having a duration of .theaorder or a tram-- tion; of. a microsecond: 'llo preserve thecharacter or; such; a pulse the; receiyer: must pass a. wide; band of; firequenciesawith. minimum attenuation; Since the: receivers: of these systems" must also;

possess gainH a relatively largen-umber of IF bandwidth in an IF amplifier, and simultaneously 7 maintaining the, gain per stage and overall amplification. It is another object of the invention to provide, means. ior. producing in an .electronic amplifier employing single-tuned circuitsa selectivitycharacteristib: halvingthemaximum possible bandwidthwhile maintaininga: single-- peaked-z upper portibn-z, It; is; a. further? object-to provide criteria by means of which an amplifier of a giitemnumberi of: stagesmay he; designed to' pass a band on frequencies having given limits.

The; invention, in general contemplates, tuning; successive single-tuned stages; of? an IF amplifier to. difierent. center. fieqll'encies. according... to

criteria determined the desired overall band width, theoverall centenf'req-uency; and? the number of amplifienstagesito-be jointlyemployed.

other objects. tenures, and advantages; of thisinvention: Willi suggest themselves to,- those skilled in the art and; will: become; apparent; from the follcwing description; of" the." invention taken in.

connectiomwith the accompanying. drawing in which: p

Fig. 1 is a schematicidiagramlottheAnCiequivalentcircuit for asingle-tunedlF amplifier stage;- and 7 Claims. (01. 15219 111 Fig. 2 is a curve showing the variation of the product of stage gain and overall bandwidth as a function of the amount of separation between the frequencies to which the two stages of a staggered pair as i Reference is now made more particularly to Fig. 1 in which there iS'ShOWH the A.-C. equivalent coupling circuit between two successive stages of a singleetunedamplifier employing. vacuum' tubes; la and: t2 respectivelm the circuit elementsrshowm representing lumped constants. Re:- sist-or M of value: It represents the: parallel combination of the plate load resistance, the plate resistance of vacuum tube ID, the input resistance of vacuum tube ILandlthe equivalent shunt resistances of the col-land capacitor of the tuned circuit. Inductance [.6 of. valuethe tuning. coil' inductance, and capacitor I'd ofiual ueflrepre sents the total circuit capacitance made of the output capacitance of vacuum tube In, the input capacitance of vacuum tube l2, and the stray wiring capacitance of the. coupling circuit.

The parallel. impedance ('Z) of this ciiicuit; which is ai'un'ct'ion or, frequency, be shown to vary according to the equation w owl 40:

where or is the. frequency expressed. in radians per second, i

1 Q=w RC, and 00 :3

r' If the transconductance of vacuum tube II] is gm,

the voltage gain (A) from the grid of this tube to the grid of vacuum tube I2 is gmR at the resonant frequency we, and

as a function of frequency.

The: complex gain function expressed by Equat1on 1i displays geometric symmetry; that is, for any two frequencies or and having we as their geometric mean, the absolute values of A are equal, and the phase angles are opposite in sign. V

In this discussion bandwidth means 3db bandwidth, i. e., the bandwidth included betweenl the two half-power or .707 voltage points on. the gain characteristic. It can be shown that the bandwidth for this single stage is imradians per second, and that the Q of the cir- 3 cuit as defined previously is equal to the ratio of center frequency to bandwidth. 7

An approximation may be made for high Q circuits; namely,

L 1+ja;

where a: is a function of frequency.

A figure of merit for a one-stage amplifier is the-product of voltage gain at band center and bandwidth. If the stage is single tuned, then since the bandwidth in cycles per second is which is independent of R andfrequency.

: When dealing with an n -stage amplifier the appropriate figure of merit is (overall gainl "x overall bandwidth If we denote (overall 'gain) by mean stage gain, the figure of merit becomes mean stage gainxoverall bandwidth 4 and this can be shown to decrease quite rapidly as n increases for synchronous single-tuned stages.

It can be shown mathematically that by tuning successive single-tuned stages of an amplifier to different frequencies or, as it will be referred to in this application. stagger tuning, one may obtain an amplifier the gain of which has an absolute value proportional to the gain factor where a: is a function of frequency as before, and

n is the number of stages. It can be shown fur-t ther that the product of meanstage gain and overall bandwidth for such. an amplifier can be kept the same as for a single stage, rather than decreasing with the number of stages, as is the case with a synchronous single-tuned amplifier as mentioned above. The selectivity curve resulting from the gain factor above has the maximum possible bandwith, if single-tuned stages are employed, while maintaining a flat top, the flat portion becoming wider as n increases.

The proper'frequenc'ies to which the various stages should be tuned and the individual stage bandwidths to obtain the results mentioned above may be computed mathematically by two methods. The choice of method depends on whether or not the high-Q approximation with arithmetic symmetry Equation 2 may be used.

The results obtained in the high Q case are summarized in Table l for various numbers of staggered stages from two (staggered pair) to 9 (staggered nonuple)." i

The results obtained in the exact case, taking into account geometric symmetry, are summarized in Table 2 for all casesfrom a'staggered pair to a staggered quintuple. The advantage of using the approximations ofTable 1 is that of much greater simplicity of computation, as can be seen by comparison to the two tables.

TABLE I.-STAGGERED 'n-UPLEs High-Q case: Arithmetic symmetry [Center-frequency=fu, overall bandwidth=Af, and f/fo small component single-tuned stages 2 staggered-pair two stages of bandwidth .71Af, staggered at fod=.35Af..

3 staggered triple.

two stages of bandwidth .5Af, staggered at fo:l:'.43A/. one stage of bandwidth A centered at In.

4 staggered-quadruple two stages of bandwidth .38Aj', staggered at fo:|:.46Af. two stages of bandwidth .92Af, staggered at fo:l:.19Af.

5 staggered-quintupla two stages of bandwidth .SlAf, staggered at faiEASAf. two stages of bandwidth .SlAf, staggered at IoiJQAI. one stage of bandwidth A f, centered at 1'0.

6 staggcred-sextuple.

two stages of bandwidth i26Af, staggered at fo=l=.48Af. two stages of bandwidth .7lAf, staggered at f0:l:.35Af. two stages of bandwidth .97Af, staggered at fo=l:.13Af.

7 staggered-septuple two stages of bandwidth ,22Af, staggered at IoiAQAf. two stages of bandwidth .62Af, staggered at fo:l:.39Af. two stages of bandwidth .QOA], staggered at f0:l:.22Af. one stage of bandwidth AL centered at-fa.

8 staggered-octuple;..

two stages of bandwidth .ZOAf, staggered at foi'AQAf. two stages of bandw dth .56Af, staggered at fu=l=.42Af. two stages of bandw dth .83Af, staggered at fo:l:.28Af. two stages of bandwidth .98Af, staggered at fo- -|;.1OAf.

9 staggered-nonuple.

.77Af, staggered at fo:|:.32Af.

mangoes;

Tents II.$1'AccaRnn n-n'ptns Exact case: Geometric symmetry component single-tuned stages staggered-pair two stages of dissipation factor d, staggered at foot and fold, where staggered-triple two stages of dissipation factor d, staggered at foa and fo/a, one stage of dissipation factor 6, centered at f0, where a 2 4 d g +6 and (a staggered-quadruple" fu/az, where two stages of dissipation factor (1!, staggered at fear and fn/a two stages of dissipation factor (1 staggered at foaiend 4 8 16 5. 1 4 rid-w and (ml- +d1 =ai staggered-quintuple two stages of dissipation factor d1, staggered at fear and a two st es of' dissi and fu/aa. One stage of dissipation factor 6, centered at uf here ag patwn factor staggered at has l 1 I: 2 and (as +113 5' 1 Dissipation factor of a single-tuned circuit=l/Q bandwidthlresonant frequency.

A plurality of staggered n-uples obtained from the information in the tables maybe serially connected just as are single stages to form a complete amplifier. Since the selectivity curves of these 'n-uples are more square-topped than those of single stages, the overall bandwidth goes down a wide IF bandpass in order to obtain noncritical tuning but is not too much concerned with maintaining a flat-topped selectivity curve. It is then possible to increase the amount of staggering, so that it exceeds the values listed in the tables and thereby realize a useful increase in the product of stage gain and overall bandwidth. Fig. 2 shows the variation of this product as a function of the amount or staggering in the case of a staggered pair; The ratio of the separa tion of the peak frequencies of the individual stages to the bandwidth of each stage is plotted along the horizontal axis, and the product of stage gain and overall bandwidthis plotted along the vertical axis. The right-hand region of the curve in which the separation ofthe individual resonance peaks exceeds the bandwidth of the individual circuits represents an over-staggered? condition. In this region the overall selectivity curve displays a double hump, which is not a seri ous deficiency in many cases. This method of overstaggering may be applied as well in the case of more than two stages, and a similar increase in the product of stage gain and overall band-- width'is obtained. v r

The equations for (11: and 6 are derived as follows: Suppose for the purpose of this derivation that a single tuned stage peaked at a frequency a is followed by a single tuned stage peaked at "l/u so that i=1 is the geometric mean of the two resonant frequencies. Suppose further that the two stages have the same Q hence the same d, The complex gain function of the combination is and now We have (III) center frequency i It can also be shown that Equation IV above is I H the absolute value of fier or a pair of amplifiers representing each term of Equation IX.

for 11 equal to an even integer, or

[swa /n sin gg cc-l/m [Hav l/m (v) for n equal to an odd integer. 15

We can now synthesize the absolute value of Equation IV by equating the coefficients of 9(f-1/f) in Equations III and V for the first factor of Equation V. For example, we need to know the dissipation factor d and the resonant frequency on satisfying d (oq+1/a sin l v11 25 squaring Equation VII and replacing (oar-l-l/afl by (a11/a) by (u11/oc) +4, we get Inserting (VI) d d (4+a 5111 %;0

Solving and finally using the double angle formula is the absolute value of By analogy to Equation VIII and I where k=1, 3, 5,

It will be recognized that Equations X and XI represent single tuned stages of dissipation factor dk and resonant in pairs at frequencies at: and 1 /ak.

While there has been described what is at present considered to be the preferred embodiment of this invention, it will be obvious to those skilled in the art that various changes and modifications may be made therein without departing from the scope of the invention as set forth in the appended claims.

The invention claimed is:

1. In a method of tuning an electronic amplifier including n serially connected stages, each of said stages being tuned to a resonant frequency by means of a single resonant circuit, the desired overall selectivity characteristic of said amplifier having a bandwidth A) between points where the absolute value of the ordinate is approximately .707 of the value of said ordinate at frequency In, the desired maximum point of said characteristic, the ratio It I being equal to a value 6, the steps including: combining a plurality of said stages into pairs, assigning to each of said pairs a number k, where k is a different value for each pair, said values being a series of successive odd integers beginning with one, assigning to each of said pairs a set of parameters di: and uk, where k is as de fined above, determining values for (lit and oak for each pair from the equations:

4+6 /16+86 cos 3+ dk=2 and adjusting the circuit constants of each stage so for n equal to an even integer, or where M and N respectively are the terms under brackets with exponents A and B respectively;

for n equal to an odd integer.

We may now synthesize Equation IX by providing a series of amplifiers in cascade, an amplithat dk is the approximate ratio of bandwidth to resonant frequency for each individual stage of each of said pairs, and so that the resonant fre-v quency of the individual stages of each of said pairs are approximately ,fOrlk and respectively. 2. In an electronic amplifier, a plurality'o'fn' 'acteristic, the ratio rel antem t e sagesya batgsseaa mage, {each or said stages being 't'iined t a resonant frequ ncy by'ifireans a single resonant circuit,

"the overall selectivity"characteristic or said *amplifier having a bandwidth A between points where the absolute value of the ordinate is approximately-7070f the value of said ordinate at frequency fit-the maximum point or said "charg1 fo orie er-safe stages neving an individual resonant frequency in and =a ratio of stage bandwidth to resonant frequency of value 6, the remainder of said stages being combined into pairs numbered 1, 3, (11- -2), 'each for "said pairshaving araine'ters dr and tick, where kis the'number a nedt'o that pair, d1:

being the apprdximat muo of bandwidth to 3. In an electronic amplifier, a plurality 01 71, serially connected stages, n being an even integer, each of said stages being tuned to a resonant frequency by means of a single resonant circuit, the overall selectivity characteristic of said amplifier having a bandwidth A) between points where the absolute value of the ordinate is approximately .707 of the value of said ordinate at frequency fo, the maximum point of said characteristic, the ratio being equal to a value 6, said stages being combined into pairs numbered 1, 3, (11-1), each of said pairs having parameters dk and ozk, where k is the number assigned to that pair, dk being the approximate ratio of bandwidth to resonant frequency for each individual stage of each of said pairs, the resonant frequencies of the individual stages of each of said pairs being approximately foozk and respectively, dk for each stage of each of said pairs being given by the relation:

4=-|6 /16+86 cos +6 2 and no: for each of said pairs being given by the relation:

1 2 ir-Z) k 4. In an electronic amplifier, a plurality of n serially connected stages, n being an odd integer,

' frequency .70, the jniaiiiriin'n point or aid t at:

7 '1'0 -each of said "stages 'being tiin'ed "to a "resonant frequency by means of a single resonant "circuit; the overall selectivity characteristic of said am-' plifier having a bandwidth A) between points where the absolute value of .the ordinate is apf pros: mately .70? ofthef value or said uang'smail as eoihpareu with unity, "dire a -said stages havin an individual bandwidth er in and a 're ti -ant rrequency liu, "the "remainder {of said stages *beingdivided into pairs, each of said pairs of stages being assigned a number k, or a difl'erent value for each pair, said values being a series of successive odd integers beginning with one, the i-ndividual bandv ridthsqof thetwo stages of each 6' said pairsbeing approiiimately equal in cycles fpensecond to the value 1 where k is the number a' ssigned to the pair to which the stage belongs, and the resonant ireatency "of the two stages of each "bf sai'ii pairs being respectively above and below rreqeaic'y ffo by an amount approximately equal in cycles per second to the value i where was tug-numberassignedtotnepair. 5. In an electronic amplifier, a plurality tr "am neven integer,

ly connectedstaigesfiitbeiii each of said "stag's being 'tiine'dto arsonant frequency by means of a single resonant circuit, the overall selectivity characteristic of said amplifier having a 'bandwidth A) between points where the absolute value of the ordinate is approximately .707 of the value of said ordinate at frequency in, the maximum point of said characteristic, the ratio being small as compared with unity, said stages being divided into pairs, each of said pairs being assigned a number k, of a different value for each pair, said values being a series of successive odd integers beginning with one, the individual bandwidths of the two stages of each of saidpairs being approximately equal in cycles per second to the value knr (sin A f where k is the number assigned to the pair to which the stage belongs, and the resonant frequencies of the two stages of each of said pairs being respectively above and below frequency f0 by an amount approximately equal in cycles per second to the value hr at) (COS where k is the number assigned to the pair.

6. An electronic amplifier including a plurality of pairs of serially connected stages, each of said stages being tuned to a resonant frequency by means of asin'gle resonant circuit, the overall selectivity characteristic of said amplifier having a bandwidth A) between points where the absolute value of the ordinate is approximately .707

1.1 (if the value of said ordinate at frequency fo,,the maximum point of said characteristic, the ratio f being equal to a value 6, each of said pairsof stages being assigned a number k, of a different value for each pair, said values being a series of successive odd integers beginning with one, each of said pairs having parameters dig and uk, where k is the number assigned to that pair, (11: being the approximate ratio of bandwidth to resonant frequency for each individual stage of each of said pairs, the resonant frequencies of the individual stages of each of said pairs being approximately foods and respectively, the total number of individual stages being n, dk for each stage of each of said pairs being given by the relation:

relation:

(1) k 7. An electronic amplifier including a plurality of pairs of serially connected stages, the total number of individual stages being n, each of said .stages being tuned to a resonant frequency by means of a single resonant circuit,. the overall selectivity characteristic of said amplifierhaving 12 a bandwidth Af between points, where the absolute value of the ordinate is approximately .707 of the value of said ordinate at frequency In, the maximum point of said characteristic, the ratio (sin Where k is the number assigned to the pair to which the stage belongs, and the resonant frequency of the two stages of each of said pairs being respectively above and below frequency In by an amount approximately equal in cycles per second to the value where k is the number assigned to the pair.

HENRY WALLMAN.

REFERENCES CITED The following references are ofv record in the file of this patent: a,

UNITED STATES PATENTS Name Date 3 Houck Dec. 12, 1922 Schienemann Dec. 10, 1940 Number 

